In this task, we address the distributed control problem for IEEE 39-Bus New England Power System frequency regulation [1]}, [2]} with the decentralized communication topology shown in Fig. REF  (a). The main goal is to optimally adjust the mechanical power input to each generator such that the phase and frequency at each bus can be restored to their nominal values after a possible perturbation. The linearized and discretized (sampling time \(\delta = 0.2\)  s) dynamics of the power system are
\(\begin{bmatrix}\Omega (k+1) \\ \omega (k+1)\end{bmatrix} &= \begin{bmatrix}I_n & \delta I_n \\ -\delta M_p^{-1} L & -\delta M_p^{-1} D + I_n\end{bmatrix} \begin{bmatrix}\Omega (k) \\ \omega (k)\end{bmatrix} + \begin{bmatrix}0_{n \times n} \\ \delta M_p^{-1}\end{bmatrix} u(k), \\y(k) &= \begin{bmatrix}I_n, 0_{n \times n}\end{bmatrix} \begin{bmatrix}\Omega (k) \\ \omega (k)\end{bmatrix},\) 
